Abstract
We investigate solutions of Witten-Dijkgraaf-E.Verlinde-H.Verlinde (WDVV) equations. The articlediscusses nonlinear equations of the third order for a function f = f(x,t)) of two independent variables x,t. Theequations of associativity reduce to the nonlinear equations of the third order for a function f = f(x,t)) whenprepotential F dependet of the metric η. In this work we consider the WDVV equation for n = 3 case with anantidiagonal metric η. The solution of some cases of hierarchy equations of associativity illustrated. Lax pairs for thesystem of three equations, that contains the equation of associativity are written to find the hierarchy of associativityequation. Using the compatibility condition are found the relations between the matrices U, V2, V1. The elements ofmatrix V2 are found with the expression of zij and independent and dependent variables for the matrix V2. Alsosolving elements of matrix V1 expressed through yij and independent and dependent variables for the matrix V1. Weaccepted that elements of matrix V0 are zero. In the physical setting the solutions of WDVV describe moduli spaceof topological conformal field theories [1, 2]. Let us introduce new variables a, b, c. In the above variables thenonlinear equations of the third order for a function f = f(x,t)) we rewritten as a new system of three equations.Expressed are variables at ,bt ,ct of three equations are written with the help of matrix elements zij ,yij.